# Functional Dependence between the Hamiltonian and the Modular Operator Associated with a Faithful Invariant State of a $W_{∗}$-Dynamical System

### J. de Cannière

University of California, Berkeley, USA

## Abstract

Let $H$ and $Δ$ be the hamiltonian, resp. the modular operator associated with an invariant faithful normal state $ω$ of a $W_{∗}$-dynamical system $(A,α)$. Then $Δ=f(h)$ for some decreasing function $f$ if and only if (roughly speaking) $ω$ is 2-passive with respect to $α$. It follows that under certain conditions a 3-passive state is an equilibrium (i.e. KMS) state.

## Cite this article

J. de Cannière, Functional Dependence between the Hamiltonian and the Modular Operator Associated with a Faithful Invariant State of a $W_{∗}$-Dynamical System. Publ. Res. Inst. Math. Sci. 20 (1984), no. 1, pp. 79–96

DOI 10.2977/PRIMS/1195181829